Digital LTI systemAnalog LTI system

DecimationSampling

DAA filterAAA filterAmplifierSeismometer

Filtering (Digital Systems)

x n[ ]

X~ k[ ]

X z[ ]

y n[ ] = h[n] *x[n]

Y~ k[ ] T~ k[ ]X~ k[ ]=

Y z[ ] T z[ ]X z[ ]=

T~ k[ ]

T z[ ]

h n[ ]

input outputfilter

Convolution of Sequences

Discrete Fourier Transform (DFT)

z-Transform

The z-Transform

Z x n[ ]{ } x n[ ]z n–

n ∞–=

∞∑ X z( )= =

T z( ) Z y n[ ]{ }Z x n[ ]{ }--------------------- Y z( )

X z( )-----------= =

Transfer Function

Properties

(shifting theorem) x n n0–[ ] zn

0–

X z( )⇔

x n–[ ] X 1 z⁄( )⇔

x1 n[ ]*x2 n[ ] x1 m[ ]x2 n m–[ ] X1 z( ) X2 z( )⋅⇔

m ∞–=

∞∑=

(convolution theorem)

T z( ) Y z( )X z( )-----------

blz l–l 0=

M

∑

akz k–k 0=

N

∑------------------------= =

Rational Transfer Function

akk 0=

N

∑ y n k–[ ] blx n l–[ ]l 0=

M

∑=

Linear Difference Equation⇔

⇔

akk 0=

N

∑ y n k–[ ] blx n l–[ ]

l 0=

M

∑=

non-recursive

FIR Filtersa0 1= ak 0= k 1≥and for( )

a0y n[ ] akk 1=

N

∑ y n k–[ ]+ blx n l–[ ]

l 0=

M

∑=

Recursive/Non-Recursive Filters

y n[ ]aka0------

k 1=

N

∑ y n k–[ ]bla0------x n l–[ ]

l 0=

M

∑+–=

y n[ ] blx n l–[ ]

l 0=

M

∑=

recursive

( IIR Filters )

T z( ) Y z( )X z( )-----------

blz l–l 0=

M

∑

akz k–k 0=

N

∑------------------------= =

Rational Transfer Function

FIR filter

y n[ ] blx n l–[ ]l 0=

M

∑ x n[ ]*b[l]= =

a0 1= ak 0= k 1≥and for( )

T z( ) blz l–l 0=

M

∑ b0 1 clz 1––( )l 1=

M

∏= =

z M– b0 z cl–( )l 1=

M

∏⋅=

Special Case:

Linear Difference Equation

• FIR filters : + Always stable. - Steep filters need many coefficients.+ Both causal and noncausal filters can be implemented. + Filters with given specifications are easy to implement!

• IIR filters : - Potentially unstable and subject to quantization errors. + Steep filters can easily be implemented with a few coefficients. Speed. - Filters with given specifications are in general, difficult, if not impossible, to implement exactly(!).

-t1 -t2

akk 0=

N

∑ y n k–[ ] blx n l–[ ]

l 0=

M

∑=

non-recursive

FIR Filtersa0 1= ak 0= k 1≥and for( )

a0y n[ ] akk 1=

N

∑ y n k–[ ]+ blx n l–[ ]

l 0=

M

∑=

Recursive/Non-Recursive Filters

y n[ ]aka0------

k 1=

N

∑ y n k–[ ]bla0------x n l–[ ]

l 0=

M

∑+–=

y n[ ] blx n l–[ ]

l 0=

M

∑=

recursive

( IIR Filters )

How can we remove these effects?

Back to Transfer function:

Frequency

Waveform Properties and Root Positions

minimum delay/phase

maximum delay/phase

mixed delay/phase(zero phase)

mixed delay/phase

mixed delay/phase

flip

time

axis

Problem: Two-Sided IRCure: Change IR into Minimum Phase

Zero Phase FIR Filter

Methods:

1) Add phase of Minimum Phase Filter to trace spectrum

2) Recursive Filtering of time inverted trace

Removing the acausal response of a Zero Phase FIR Filter

Linear PhaseFIR filter

zlp

X z( ) Y z( )

F z( ) time delay lp

00

Linear PhaseFIR filter

X z( )Y z( )

F z( )

lp00 0

linear phaseshift correction

Linear Phase and Zero Phase Filter:

Linear-phase FIR Filter:

Zero-phase FIR Filter:

= Time delay correction by samples

F z( )

F z( ) F z( ) zlp⋅=

zlp lp

General rule:Any filter can be expressed by convolution of its minimum phase and maximum phase component.

roots within UC roots outside UC

=> Linear phase FIR Filter: F z( ) Fmax z( ) Fmin z( )⋅=

maximum phase component -> left-sided (acausal) componentminimum phase component -> right-sided (causal) component

Removal of acausal response:

Replace maximum phase component Fmax(z) by its minimum phase equivalent MinPhase{Fmax(z)}.

MinPh ase Fmax z( ){ } =

Answer: flip maximum phase component in time. In terms of z-transform:

=> MinPhase Fmax z( ){ } Fmax 1 z⁄( )=

minimum phase

maximum phase

z-transform representation of digital seismogram

Y~ z( ) F z( ) zlp X~ z( )⋅ ⋅=

digital seismogram before FIR filtering

digital seismogramafter FIR filtering but before decimation

LP FIR

0P FIR

LP correction

x~ n[ ] = unfiltered digital seismogram~X z( ) = z-transform of the input signal x~ n[ ]

F z( ) = z-transform of the linear phase FIR filter

y~ n[ ] = filtered digital seismic trace before decimation1

1 This is a fictitious signal since decimation is commonly done while filtering.

= z-transform of Y~ z( ) y~ n[ ]

F z( ) zlp⋅corresponds to a zero phase filter in which the linear phase component of F(z) is corrected.

In practice: Treat time shift zlp separately from F(z)

Removing the maximum phase component of a FIR filter

Principle:

Fmax z( ) -> MinPh ase Fmax z( ){ }

‘Corrected’ seismogram Y(z): Y z( ) 1Fmax z( )-------------------- Fmax 1 z⁄( ) Y

~z( )⋅ ⋅=

Problem: Since Fmax(z) has only zeros outside the unit circle, 1/ Fmax(z) will have poles outside the unit circle.

Solution: Flip time axis.

Y 1 z⁄( ) 1Fmax 1 z⁄( )--------------------------- Fmax z( ) Y~ 1 z⁄( )⋅ ⋅=

Impulse response corresponding to 1/ Fmax(z) becomes a stable causal sequence in nominal time and the deconvolution of the maximum phase component Fmax(z) poses no stability problems.

The difference equation

Fmax 1 z⁄( ) Y 1 z⁄( )⋅ Fmax z( ) Y~ 1 z⁄( )⋅=

Rewrite to

A' z( ) Y ' z( )⋅ B' z( ) X ' z( )⋅=

Written as convolution sum:

A' z( )

<=>

Fmax 1 z⁄( ) Y' z( ) Y 1 z⁄( )

B' z( )

<=>

Fmax z( ) X' z( ) Y~ 1 z⁄( )

<=>

<=>

a' k[ ] y' i k–[ ]⋅k ∞–=

∞

∑ b' l[ ] x' i l–[ ]⋅l ∞–=

∞

∑=

Assumption: F(z) contains mx zeros outside the unit circle=> wavelets a’[k] and b’[k] will be of length mx + 1.

a' k[ ] y' i k–[ ]⋅

k 0=

mx

∑ b' l[ ] x' i l–[ ]⋅

l 0=

mx

∑=

Rearrange to

y' i[ ] a' 0[ ]⋅ a' k[ ] y' i k–[ ]⋅

k 1=

mx

∑+ b' l[ ] x' i l–[ ]⋅

l 0=

mx

∑=

which is equivalent to

y' i[ ] a' k[ ]a' 0[ ]------------ y' i k–[ ]⋅

k 1=

mx

∑– b' l[ ]a' 0[ ]------------ x' i l–[ ]⋅

l 0=

mx

∑+=

This is

for k = 1 to mxa k[ ]a' k[ ]a' 0[ ]------------–

fmax mx k–[ ]

fmax mx[ ]--------------------------------= =

and

for l = 0 to mxb l[ ] b' l[ ]a' 0[ ]------------

fmax l[ ]fmax mx[ ]-----------------------= =

The convolution sum

y' i[ ] a k[ ] y' i k–[ ]⋅k 1=

mx

∑ b l[ ] x' i l–[ ]⋅l 0=

mx

∑+=

y' i[ ] = the time reversed ‘corrected’ sequence.

To obtain y[i] flip y’[i] back in time.

‘Corrected’ seismogram: Y z( ) F~

z( ) Y~

z( )⋅=

‘Correction’Y~z( ) Y z( )

F~z( )

lp0

filter

lp0

F~ z( ) = changes the linear phase filter F(z) into a minimum phase filter

=> onset of output signal is advanced by lp samples.

To account for this time shift in the corrected seismogram:

a) change time tag of ‘corrected’ trace

or

b) delay ‘corrected’ trace by lp samples.

What is needed for ‘correction’?

mx + 1 coefficients of the maximum phase portion of the a linear phase FIR filter

How to get maximum phase component?

F z( ) b lz l–

l 0=

m

∑ b0 1 c lz l––( )l 1=

m

∏= =

mn zeros inside UC (cimin) <=> Fmin(z)

mx zeros outside UC (cimax) <=> Fmax(z)

Fmax z( ) fimaxz i–

i 0=

mx

∑ b0 1 cimaxz i––( )

i 1=

mx

∏= =

From zeros ouside UC -> polynomial Fmax(z). fmax[l]for l = 0 to mx: coefficients of polynomial Fmax(z).

Correction Procedure

‘decimation while filtering’

x n[ ]y n[ ]

digitalsignal

Fs :f1decimateddigitalsignal

Stage 1f1:f2

Stage 2fn 1– :fn

Stage n...

Fs

f1 f2

fnintermediate data streams

initial samplingfrequency

final sampling frequency

Full Correction (not always necessary):

interpolation -> fn 1– -> correction for FIR filter stage n

interpolation -> fn 2– -> correction for FIR filter stage n-1

interpolation -> f1 -> correction for FIR filter stage 2interpolation -> Fs -> correction for FIR filter stage 1

finally: decimation back to fn

. . .

.

Decimation stages

a) b)

a) b)

1 423

13

515

14

16 17

12

7 8109

11

6 1 2

34

5

1514

17

16

12

611

78

109

13

Conclusions

FIR filter generated precursory artefacts:

• can become impossible to be identified visually

• can have similar scaling properties as nucleation phases

Zero - phase FIR filters in general

• affect the determination of all onset properties (onset times, onset polarities)

For the interpretation of onset properties (onset times, onset polarities, nucleation phases, etc.) the acausal response of thezero-phase FIR filter has to be removed

but not

for waveform analysis.

Consequence